Question: Simplify the following expression and state the condition under which the simplification is valid: $r = \dfrac{p^2 + 12p + 20}{p^2 + 10p}$
First factor the expressions in the numerator and denominator. $ \dfrac{p^2 + 12p + 20}{p^2 + 10p} = \dfrac{(p + 2)(p + 10)}{(p)(p + 10)} $ Notice that the term $(p + 10)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(p + 10)$ gives: $r = \dfrac{p + 2}{p}$ Since we divided by $(p + 10)$, $p \neq -10$. $r = \dfrac{p + 2}{p}; \space p \neq -10$